In this section we present and prove two lemmas, Lemmas REF  and REF , which extend some earlier results on bridge-addable sets of graphs, see [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}. We shall use these new results in the proof of Lemma REF , which is used in the proof of Theorem REF , and elsewhere in the proof of Theorem REF . (The other lemma in this section, Lemma REF , is used only in the proof of Lemma REF .)
We need some definitions.
